Since both statements are statements of real numbers, the predication of truth is a function of axioms, theorems and proof of reals; establishing whether or not the veracity of a mathematical statement is false, conditionally true, or universally true is in essence one of the major functions of mathematical proof. Hence, when an algebraic statement is true, we know because we prove (direct proof, for example) or disprove (proof by contradiction, for example).
In the first case, we know the claim
2x+10=20 is conditionally true because we use a series of algebraic axioms to reduce the statement to
x=5 [particularly by inverting the operations and their order such that if our original statement is some quantity subjected to operations
(*2,+10,=20), then it is true in any cases residual of applying operations
(=20,-10,/2)]. So, we do the algebra and see what we got. In fact, the two statements combined can be shown consistent because they instantiate the identity axiom
20=20. Thus, the truth of the statement S1(x) is conditional on x.
We have exactly the same challenge in the second, and instead find by simplifying the RHS to
2x=2x. No matter how hard we try to simplify the equation, we will not be able to isolate the variable. As such, any value works by direct proof, again invoking the identity axiom
a=a which we presume to be true. (It would be messy otherwise.) Thus, the statement S2(x) is tautological.
Let's do a third example:
x + x + x = 2x. Here, if we simplify and then divide both sides by quantity (barring 0 as a value), it reduces to 3=2. This violates the identity axiom
a=a. So, since
a=a we must reject either the axiom or the conclusion, and by extension the original statement. Thus, while we can create the appearance of a predication of some quantity of the reals, using the rules that govern reals, we find the predication is always false. Thus, the predication S3(x) is contradictory and necessarily false (if we want to hold on to our axioms).
Lastly, the truth value of a predication may not be possible to be proven within the axiomatic formal system from which it arises. A famous example is the Goldbach Conjecture which states that:
every even whole number greater than 2 is the sum of two prime numbers. The conjecture has been shown to hold for all integers less than 4 × 10^18, but remains unproven despite considerable effort.
So, S4(n,p1,p2) which is
2n=p1+p2 for n in naturals, and p1 and p2 in primes seems to be true, but no proof exists for the predication, and no counterexample has arisen either. Thus, it is unknown whether this predication is conditional or tautological.
Don't let the philosophical jargon predication complicate your understanding. In mathematics, predication is nothing more than binding a variable to a domain and then predicating the truth-condition on the consistency of the statements within an axiomatic system, and I suspect you've been doing it by at least intuition since elementary school. We know when mathematical predications are true when we devise mathematical proofs, disproofs (as in the irrationality of 2), or when we discover counterexamples. And ultimately to become a professional mathematician is to fully explore both proof theory and model theory.